Over the last couple of years American Football gained a lot of popularity in Europe especially Germany. Different than soccer, American football is a play based game. After each play the game gets interupted for the offense and defense to set up their team once again. The goal is to achieve a touchdown on the opponents endzone. To achieve that the offense works its way across the field play by play. With each play the offense tries to gain as many yards as possible while the defense tries to prevent this. There are 2 main kind of plays to gain yards: rushing and passing. For a rush plays the offensive team tries to gain yards by caring the ball. For pass plays the quarterback of the offensive team will throw to ball to another player. The key difference is the that for rushing plays the ball will not be airborn but handed over.
We want to find out if there is a way to predict how many yards a team will gain in a play based on variables that we know before the play starts.
Other people have already researched similiar questions but i.e. focused on predicting the rushing yards. We focus on yards generally gained no matter if rushed or passed, as this can not be determined before the play starts. Interesting links to check out for similiar analysis:
For NFL matches there is a comprehensive data asset available that includes play-by-play data, we will focus on the data gathered so far (until 26th December 2022) in the 2022 NFL season. It tracks a multitude of variables for each play. For our analysis we will focus on the variables that are known before the play starts as explanatory variables and the yards gained as the response variable. The data was scraped from the NFL Next Gen Stats website with the NFL verse package.
Data Source https://github.com/nflverse/nflverse-data/releases/tag/pbp
Data Description https://mrcaseb.github.io/pages_dummy/reference/fast_scraper.html
Our response variable will be “yards-gained” this variable describes how many yards a team gained (or lost) in a play excluding yards gained via fumble recoveries and laterals. Our possible explanatory variables can be found in the data dictionary below.
Data Dictionary
Show the code
#Import librariesimport pandas as pdimport data_dic #Create Data Framedata_dictionary_content = data_dic.data_dictionary_contentdata_dictionary = pd.DataFrame(data_dictionary_content)data_dictionary
As we want to find out if there is a way to predict how many yards a team will gain based on variables that we know before the play starts, we reduce the variables to the following small list which represents the variables, that are already known before a play. We also drop all plays during which one of the variables is not available.
Show the code
# Select the relevant variablescolumn_selection = ["yards_gained","home_team","away_team","quarter_seconds_remaining","qtr","down","yardline_100","ydstogo","posteam","defteam","score_differential"]# Filter dataframe for relevant variablesdf_small = df[column_selection]# drop rows with missing valuesdf_small.dropna(inplace=True)
We differentiate between categorial variables and numerical variables.
Show the code
# List of categorial predictive variableslist_categorial = ["posteam","defteam","home_team","away_team"]# List of our numeric predictive variableslist_numeric = ["quarter_seconds_remaining","qtr","down","yardline_100","ydstogo","score_differential"]
Data preparation
Next we split our data into explanatory variables and the response variables.
Show the code
# define outcome variable as y_labely_label ='yards_gained'# select features and drop y-variableX = df_small.drop(['yards_gained'], axis=1)# create responsey = df_small[y_label]
For the categorial variables we need to create dummy columns to fit them into our regression models.
Show the code
# Data correction for the categorial variablesX[list_categorial] = X[list_categorial].astype("category",copy=False)# Create dummies for the categorial variables, so we can fit them in our modeldummies = pd.get_dummies(X[list_categorial])# convert the headers to lower_casedummies.columns = [x.lower() for x in dummies.columns]# concatenate the categorial variables to the numerics and get all predictive variables in X.X = pd.concat([X[list_numeric], dummies], axis=1)
Data splitting
We do a train/test split with 70/30 % of the data.
We notice that the mean of yards gained after each play is about 4.6 with standard deviation of 8.1
The rest of the data only reflects the rules and characteristics of the game.
Exploratory data analysis
Show the code
# Histogram occurency of Yards gaineddf_tmp = pd.DataFrame(df_small).copy()alt.Chart(df_tmp).mark_bar().encode( x=alt.X("yards_gained"), y=alt.Y('count()'),).interactive()
We can see that many plays result in zero yards gained. This is due to plays being aborted i.e. by incomplete passes or fouls.
Show the code
# Histogram occurency of Yards gained without plays were no yards were gained/lostdf_tmp=df_small.loc[df.yards_gained !=0]alt.Chart(df_tmp).mark_bar().encode( x=alt.X("yards_gained"), y=alt.Y('count()'),).interactive()
After filtering out the plays that result zero yards gained, we get a distribution, that looks like a positive skewed normal distribution, with a mean = 4.6.
Show the code
# Calculation of probability that a team gains yards with a playprint("Probability that a team gains positive yards with a play: "+str(round((len(df_small.loc[df_small.yards_gained >0]) /len(df_small)) *100, 2) ))
Probability that a team gains positive yards with a play: 57.94
Probability that a team gains positive yards with a play is 57.94 %
Relationships
Now we want to understand the relationship between our numerical variables.
Show the code
# Explorative analysis of correlation between the variablesalt.Chart(df_tmp).mark_circle().encode( x=alt.X(alt.repeat("column"), type='quantitative', scale=alt.Scale(zero=False) ), y=alt.Y(alt.repeat("row"), type='quantitative', scale=alt.Scale(zero=False) )).properties( width=150, height=150).repeat( row = list_numeric + ["yards_gained"], column=list_numeric + ["yards_gained"])
Some relationships in the data can already be seen, most of them are game roles and restriction related i.e. - the maximum yards that can be gained are to the finish line, therefor there is a hard cut off and yards_gained can never be bigger than yardline_100 - a high score_differntial is uncommon therefor some relationships look like a normal distribution i.e. yards_gained & score_differential this is probably due to the fact, that high yard gains are uncommon as well as high score differences, therefor this situations don’t occure often - the score_differential becomes wider with every quarter, which makes sense, as the dominant team has more time to score points - the 5th qtr is overtime and therefor shorter, which also leaves less occassions for potential high yard plays. As can be seen the yards_gained and qtr relationship - it seems in the 4th down there are less yards_gained then in the others. Which can be explained by team going for a kick in the 4th down to put the ball deep into the opponents half but does not gain any yards.
Correlation Matrix
With a correlation matrix we want to test which of our variables correlate to each other by what amount. For that we chose the Spearman Method over the Pearson and Kendall Method. It does not really on normality of the data as the Pearson method does. And tests for correlation not dependence as the Kendall Method does
We can see a that most of the variables do not strongly correlate with each other.
ydstogo and down are quite strongly correlated as expected, because with each down the offensive team should come closer to the yardline it needs to reach.
ydstogo and yardline_100 also have some correlation which can be explained by the underlying rules of the game. The offensive team can not have more yards to go than they have until they enter the defenses endzone, the distance to which is represented by the yardline_100 variable. Therefor it’s always ydstogo <= yardline_100
yards_gained and down also seem to have a correlation which can be explained by - the offensive team rather punting (kicking the ball to the direction of the defenses endzone which usually allows the defense to become offense) than trying to gain yards on 4th down - game tactics as the target distance ydstogo correlates with down. Teams might more often go for plays that result in just making it to the yardline on 3rd or 4th down rather than aiming for big plays that result in many yards gained.
Modeling
In this section we will look at 3 different regression models (Linear regression, K-nearest neighbour, & Lasso-regression) as well as Model 4 were we just always predict the mean value as a sanity check.
Model 1: Linear regression model
Show the code
# Select a linear regression modelreg = LinearRegression()X.head()
quarter_seconds_remaining
qtr
down
yardline_100
ydstogo
score_differential
posteam_ari
posteam_atl
posteam_bal
posteam_buf
...
away_team_no
away_team_nyg
away_team_nyj
away_team_phi
away_team_pit
away_team_sea
away_team_sf
away_team_tb
away_team_ten
away_team_was
2
896
1
1.0
78.0
10
0.0
0
0
0
0
...
0
0
0
0
0
0
0
0
0
0
3
869
1
1.0
59.0
10
0.0
0
0
0
0
...
0
0
0
0
0
0
0
0
0
0
4
865
1
2.0
59.0
10
0.0
0
0
0
0
...
0
0
0
0
0
0
0
0
0
0
5
841
1
3.0
54.0
5
0.0
0
0
0
0
...
0
0
0
0
0
0
0
0
0
0
6
833
1
4.0
64.0
15
0.0
0
0
0
0
...
0
0
0
0
0
0
0
0
0
0
5 rows × 134 columns
Forward selection
Because we have a lot a variables, we would like to choose the 10 most relevant variables for the model to make it more sensitive in regards to these important variables. Doing so, we use Forward selection:
# We redefine our predictive data and select only the relevant 10 variablesX.columns[sfs_backward.get_support()]relevant_features = X.columns[sfs_backward.get_support()]X_train = X_train[relevant_features]X_test = X_test[relevant_features]
Show the code
# Fit the modelreg.fit(X_train, y_train)
LinearRegression()
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
LinearRegression()
After the model training we can check the 10 most important variables and their coefficient.
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
KNeighborsRegressor(n_neighbors=2)
Evaluation on Test set
Show the code
y_pred2 = reg2.predict(X_test)
Show the code
# Mean squared error model 2model2MSE = mean_squared_error(y_test, y_pred2)model2MSE
86.06705172085948
Show the code
# Root mean squared error model 2model2RMSE = mean_squared_error(y_test, y_pred2, squared=False)model2RMSE
9.277232977610268
Show the code
# Mean absolute error model 2model2MAE = mean_absolute_error(y_test, y_pred2)model2MAE
6.069832667807568
Save Model
Show the code
ts = time.time()file='../models/KN_Model.sav'+ datetime.datetime.fromtimestamp(ts).strftime('_%Y-%m-%d-%H_%M_%S') +".sav"# pickle.dump(reg2, open(file, 'wb')) # uncommented to reduce dumping models, after each reload. file
'../models/KN_Model.sav_2023-01-16-21_38_43.sav'
Model 3: Lasso Regression
Standardization
Lasso performs best when all numerical features are centered around 0 and have variance in the same order.
If a feature has a variance that is orders of magnitude larger than others, it might dominate the objective function and make the estimator unable to learn from other features correctly as expected.
This means it is important to standardize our features.
# numpy.linspace(start, stop, num of samples)alphas = np.linspace(0.01,500,100)lasso = Lasso(max_iter=10000)coefs = []for a in alphas: lasso.set_params(alpha=a) lasso.fit(X_train, y_train) coefs.append(lasso.coef_)
Show the code
ax = plt.gca()ax.plot(alphas, coefs)ax.set_xscale('log')plt.axis('tight')plt.xlabel('alpha')plt.ylabel('Standardized Coefficients')plt.title('Lasso coefficients as a function of alpha');
Lasso with optimal alpha
To find the optimal value of alpha, we use scikit learns lasso linear model with iterative fitting along a regularization path (LassoCV).
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
LassoCV(cv=5, max_iter=10000, random_state=0)
Show the code
# Show best value of penalization chosen by cross validation:reg.alpha_
0.0009685368136113017
Show the code
# Set best alphalasso_best = Lasso(alpha=reg.alpha_)lasso_best.fit(X_train, y_train)
Lasso(alpha=0.0009685368136113017)
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
Lasso(alpha=0.0009685368136113017)
Evaluation on Test set
Show the code
# Show scoreprint('R squared training set', round(lasso_best.score(X_train, y_train)*100, 2))print('R squared test set', round(lasso_best.score(X_test, y_test)*100, 2))
R squared training set 2.55
R squared test set 2.28
Show the code
# Mean squared error model 3model3MSE = mean_squared_error(y_test, lasso_best.predict(X_test))model3MSE
63.43151846072908
Show the code
# Root mean squared error model 3model3RMSE = mean_squared_error(y_test, lasso_best.predict(X_test), squared=False)model3RMSE
7.964390652192362
Show the code
# Mean absolute error model 3model3MAE = mean_absolute_error(y_test, lasso_best.predict(X_test))model3MAE
5.307091618975074
Show the code
plt.semilogx(reg.alphas_, reg.mse_path_, ":")plt.plot( reg.alphas_ , reg.mse_path_.mean(axis=-1),"k", label="Average across the folds", linewidth=2,)plt.axvline( reg.alpha_, linestyle="--", color="k", label="alpha: CV estimate")plt.legend()plt.xlabel("alphas")plt.ylabel("Mean square error")plt.title("Mean square error on each fold")plt.axis("tight")ymin, ymax =60, 80plt.ylim(ymin, ymax);
Show the code
# saving the lasso reg modelts = time.time()file='../models/lasso_reg_model.sav'+ datetime.datetime.fromtimestamp(ts).strftime('_%Y-%m-%d-%H_%M_%S') +".sav"# pickle.dump(lasso_best, open(file, 'wb')) # uncommented to reduce dumping models, after each reload. file
# Mean squared error model 4model4MSE = mean_squared_error(y_test, y_prediction_model4)model4MSE
64.91328211074314
Show the code
# Root mean squared error model 4model4RMSE = mean_squared_error(y_test, y_prediction_model4, squared=False)model4RMSE
8.056877938180715
Show the code
# Mean absolute error model 4model4MAE = mean_absolute_error(y_test, y_prediction_model4)model4MAE
5.451782561442774
Conclusions
Out of the 3 Models we evaluated only Linear regression and Lasso regression were able to beat our naive Model where we always predicted the mean value of the training set slightly.
Both Linear Regression and Lasso Regression performed essentially the same and only slightly better than the Dummy Model.
It seems that with the given variables we were not able to create a model which can predict yards gained in a football game reliable. This reflects the unpredictable nature of the game, which makes it enjoyable by millions of fans.
